Marcus equation derivation

The classical Marcus equation, derived from the basic model by the simplest treatment, in which all motions are treated by classical mechanics, is Equation (9.46), in which AG is the theoretical activation free energy and AG is the free energy of reaction ... 

The semi-classical Marcus equation derives from quantum-mechanical treatments of the Marcus model, which consider in wave-mechanical terms the overlap of electronic wave-functions in the donor-acceptor system, and the effects of this overlap on electronic and nuclear motions (see Section 9.1.2.8 above). Such treatments are essential for a satisfactory theory of D-A systems in which the interaction between the reactant and product free-energy profiles is relatively weak, such as non-adiabatic reactions. A full quantum-mechanical treatment, unfortunately, is cumbrous and (since the wave-functions are not accurately known) difficult to relate to experimental measurements but one can usefully test equations based on simplified versions. In a well-known treatment of this type, leading to the semi-classical Marcus equation introduced in Section 9.1.2.8, the vibrational motions of the atomic nuclei in the reactant molecule (as well as the motions of the transferring electron) are treated wave-mechanically, while the solvent vibrations (usually of low frequency) are treated classically. The resulting equation, already quoted (Equation (9.25)), is identical in form with the classical equation (9.16) (Section 9.1.2.5), except that the factor... 

For a system describable with a single progress variable, we derived the Marcus equation, Eq. (5-76). 

Actually the assumptions can be made even more general. The energy as a function of the reaction coordinate can always be decomposed into an intrinsic term, which is symmetric with respect to jc = 1 /2, and a thermodynamic contribution, which is antisymmetric. Denoting these two energy functions h2 and /zi, it can be shown that the Marcus equation can be derived from the square condition, /z2 = h. The intrinsic and thermodynamic parts do not have to be parabolas and linear functions, as in Figure 15.28 they can be any type of function. As long as the intrinsic part is the square of the thermodynamic part, the Marcus equation is recovered. The idea can be taken one step further. The /i2 function can always be expanded in a power series of even powers of hi, i.e. /z2 = C2h + C4/z. The exact values of the c-coefficients only influence the... 

Again this averaging procedure can only be expected to work when the reactions are sufficiently similar . This is difficult to quantify a priori. The Marcus equation is therefore more a conceptual tool for explaining trends, than for deriving quantitative results. 

In the previous section we have shown that the Marcus equation can be derived from Eq. (3.40). In this section, other forms of rate constants used in literatures will be derived. Notice that at T = 0, Eq. (3.40) reduces to... 

The first attempt to describe the dynamics of dissociative electron transfer started with the derivation from existing thermochemical data of the standard potential for the dissociative electron transfer reaction, rx r.+x-,12 14 with application of the Butler-Volmer law for electrochemical reactions12 and of the Marcus quadratic equation for a series of homogeneous reactions.1314 Application of the Marcus-Hush model to dissociative electron transfers had little basis in electron transfer theory (the same is true for applications to proton transfer or SN2 reactions). Thus, there was no real justification for the application of the Marcus equation and the contribution of bond breaking to the intrinsic barrier was not established. 

The equation for kobs derived by Marcus (equation 18) is essentially equivalent to the quantum mechanical result in the classical limit (equation 30). If equation (23) is used for A0 assuming a dielectric continuum, the quantum mechanical result is given by equation (40) where 4AG —... 

The Marcus equation can be derived by assuming strong vibronic coupling (J2jSj 1) and high temperature (h,Mj/knT 1). Thus, to the second-order approximation one obtains the Marcus equation ... 

In photo-induced ET the Marcus equation [58-60] is often used and it can be derived from Eq. (96) by using the short-time approximation, i.e.,... 

Derive Marcus equation from Eq. (97) using the short-time approximation and classical limit. 

An alternative way of deriving the Marcus equation is again to assume a reaction... 

These considerations were expressed quantitatively by Marcus. He derived an equation for predicting the rate constant for an outer-sphere reaction from the exchange rate constants for each of the redox couples involved and the equilibrium constant for the overall reaction. The Marcus equation for the rate constant k 2 is... 

Again this averaging procedure can only be expected to work when the reactions are sufficiently similar . This is difficult to quantify a priori. The Marcus equation is therefore more a conceptual tool for explaiiung trends, than for deriving quantitative results.-----------------------------------------------------------------------------------------... 

Semiempirical calculations of free energies and enthalpies of hydration derived from an electrostatic model of ions with a noble gas structure have been applied to the ter-valent actinide ions. A primary hydration number for the actinides was determined by correlating the experimental enthalpy data for plutonium(iii) with the model. The thermodynamic data for actinide metals and their oxides from thorium to curium has been assessed. The thermodynamic data for the substoicheiometric dioxides at high temperatures has been used to consider the relative stabilities of valence states lower than four and subsequently examine the stability requirements for the sesquioxides and monoxides. Sequential thermodynamic trends in the gaseous metals, monoxides, and dioxides were examined and compared with those of the lanthanides. A study of the rates of actinide oxidation-reduction reactions showed that, contrary to previous reports, the Marcus equation ... 

The Marcus equation (and useful relationships derived from it) is a special case characterized by adiabatic electron transfer at the intersection of the reactant and... 

Lewis and co-workers also are concerned with the reactivity-selectivity postulate (RSP), which can also be derived from the Marcus expression. In the examples given here, selectivity does not vary with reactivity, in apparent contradiction to Marcus theory. This result can be explained on the basis that the intrinsic barriers are not constant and by assuming that the quadratic term of the Marcus equation contributes very little when the identity barriers are high (as they are when rates are well below diffusion control). Other important contributions to understanding the RSP have been made recently (9a, 9b). 

Many reactions exhibit effects of thermodynamics on reaction rates. Embodied in the Bell-Evans-Polanyi principle and extended and modified by many critical chemists in a variety of interesting ways, the idea can be expressed quantitatively in its simplest form as the Marcus theory (15-18). Murdoch (19) showed some time ago how the Marcus equation can be derived from simple concepts based on the Hammond-Leffler postulate (20-22). Further, in this context, the equation is expected to be applicable to a wide range of reactions rather than only the electron-transfer processes for which it was originally developed and is generally used. Other more elaborate theories may be more correct (for instance, in terms of the physical aspects of the assumptions involving continuity). For the present, our discussion is in terms of Marcus theory, in part because of its simplicity and clear presentation of concepts and in part because our data are not sufficiently reliable to choose anything else. We do have sufficient data to show that Marcus theory cannot explain all of the results, but we view these deviations as fairly minor. 

The Br0nsted equation, which is empirical, can be viewed as being a special case of the Marcus equation, which was derived from first principles. The latter theory has the important added feature of defining intrinsic activation barriers. 

Marcus-type considerations do not alter the simple interpretation of j8 equations for 8 derived from various models [28] predict a variation in accord with the Hammond postulate. Thus the Marcus equation gives P = 0.5 when there is zero driving force for the reaction AG = 0) for an increasingly unfavourable driving force the value of increases while with a favourable one it decreases (see Eqn. 33). 

Eor ferrocene sites at the end of long alkanethiols self-organized at gold electrodes and diluted with unsubstituted thiols with the redox moiety in contact with the electrolyte (Fig. 4a), Chidsey has reported [34] curved Tafel plots (Fig. 4b), which could be fitted by equations derived from Marcus theory with values of k = 0.85 eV and Z = 6.73 x 10 s"l eV" for a reaction rate of A = 2.5 s at in Fig. 4(b). Similar curvature in Tafel plots has been reported by Faulkner and coworkers [35] for adsorbed osmium complexes at ultramicro-electrodes (UME). The temperature dependence of the rate coefficient could also be fitted from Marcus equation and electron states in the metal and coupling factors given by quantum mechanics. 

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